In September 2009, Australian astronomer Robert H. McNaught discovered comet C/2009 R1 (McNaught). The orbit of this comet is hyperbolic with the Sun at one focus. Because the orbit is not elliptical, the comet will not be captured by the Sun's gravitational pull and instead will pass by the Sun only once. The comet reached perihelion on July 2, 2010. (Source: Minor Planet Center, http://minorplanetcenter.net/) The path of the comet can be modeled by the equation (1191.2) (30.9) where x and y are measured in AU (astronomical units). a. Determine the distance (in AU) at perihelion. Round to 1 decimal place. b. Using the rounded value from part (a), if 1 AU = 93,000,000 mi, find the distance in miles. Perihelion distance (0, 0) "Sun

The orbit of Halley's Comet is an ellipse whose major axis is 3.34 × 10 ^ 9 miles long, and whose minor axis is 8.5 × 10 ^ 8 miles long. What is the eccentricity of the comet's orbit?

In a distance star system a planet is 9 AU (astronomical unit) away from the sun, and has a circular orbit with period 12 earth years. The planet has a moon distanced 0.5 AU away, which circles the planet in a circular orbit also with period 12 earth years. Suppose the orbital planes of the planet around the sun and the moon around the planet coincide, and their motions are both counterclockwise. At midnight March 1 2021, the positions of the sun, planet, and moon are observed to be (0, 0), (9, 0), (9, 0.5) on their common orbital plane. (a)  Determine the position (xP (t), yP (t)) of the planet, t years after midnight March 1 2021. (b)  Determine the position (xM (t), yM (t)) of the moon, t years after midnight March 1 2021. Express your answer in the form xM (t) = A cos(kt + α) + xP (t) yM (t) = A sin(kt + α) + yP (t) where A,k > 0 and α ∈ (−π,π]. (c)  Simplify your expressions for xM (t) and yM (t) to the form xM (t) = B cos(kt + β) yM (t) = C sin(kt + γ) where B, C, k >…

Given the equation of a conic in the form 8x^2 -12xy +17y^2 -20=0 we can convert this into standard form in the XY coordinate system without the XY term. After finding out what cot(2θ) is, we can use this information to find what sin θ and cos θ is by using the trigonometric identities 2sin^2θ=1-cos2θ and 2cos^2θ=1 +cos 2θ. So, what is the value of sin θ and cos θ?